Question

A ) Let L: R^10 --> R^10 be a bijective transformation. The Ker(L) is _________ ?

If this is not enough information to determine the Kernal, explain why.

B) Let L: R^6 --> R^6 be a surkective transformation. The (dim(Ker(L))) is _________?


E) Let A be a 4 x 4 matrix with a non-zero determinant. Then the columns of A form a basis for ________ ?

Answer 1

Explanation:

A) The Kernel of a transformation L is the set of vectors in the domain of L that get mapped to the zero vector in the codomain of L. In this case, since L is a bijective transformation, it means that L is both injective (one-to-one) and surjective (onto).

For a bijective transformation, the Kernel is always the zero vector, denoted as {0} or the trivial subspace.

B) The dimension of the Kernel, denoted as dim(Ker(L)), is the number of linearly independent vectors in the Kernel of the transformation L.

In this case, since L is a surjective transformation, it means that every vector in the codomain of L is mapped from some vector in the domain of L. Therefore, the Kernel of L is the zero vector {0}, and the dimension of the Kernel is 0.

E) The columns of a matrix A form a basis for the column space of A. The column space of a matrix is the subspace spanned by its column vectors.

In this case, since the matrix A has a non-zero determinant, it means that its columns are linearly independent, and they span the entire column space. Therefore, the columns of A form a basis for the column space of A.

To summarize:

A) The Kernel of the transformation L is the zero vector, denoted as {0} or the trivial subspace.

B) The dimension of the Kernel, dim(Ker(L)), is 0.

E) The columns of the matrix A form a basis for the column space of A.